I would not say that the need to pay for a lawyer has any effect on the desire to achieve equal justice under the law. However, it clearly does have an impact on people's ability to get equal justice.
Perhaps the most obvious example of this could be seen in the OJ Simpson murder trial from the mid-1990s. This was a case in which any defendant who could not afford a whole team of high-priced lawyers would have been convicted. (I am not saying Simpson was guilty, just that he looked guilty and would likely have been convicted without such a great legal team.) Because Simpson had money, he was able to get acquitted where a poorer defendant would surely have been convicted.
Some lawyers will do some work on a pro bono basis, but they surely cannot provide enough legal services to give all poor people the kind of legal representation that Simpson and other rich people can get.
This shows how the need to pay for lawyers affects the ability to get equal justice under the law.
There is a common conception that public defenders, legal aid attorneys, etc. are no substitute for "hired guns." This is not entirely true. Quite often, attorneys working pro bono will have substantially more expertise dealing with difficult cases than a retained attorney. Still, there are issues. Most pro bono attorneys are severely overworked. Although they are as accountable as any other attorney, they seldom can devote all their time to a single client. But for that matter, very few attorneys can. The "dream team" that O.J. Simpson retained--successfully--has been mentioned previously. However, this is the exception rather than the rule. Only someone with Simpson's wealth could retain attorneys such as this. Attorneys, like physicians, accountants, etc. either retained or assigned may be capable or incompetent. Whether they are paid or working on the public dole is not a good indicator of the attorney's competency.
While it is true that almost everyone desires a fair trial, the rich are more likely to expect it. Many poor people do not expect to get a fair trial. They understand that public defenders are overworked and often inexperienced in more serious criminal matters. They know that getting a fair trial is a long shot, but there is nothing they can do about it. The rich, on the other hand, expect to be acquitted. Sometimes they believe that they can buy justice.
I have to say that I also believe the ability to pay for a defense attorney, or team of lawyers, does have an impact on the ability to ensure justice. The Simpson case is a great example of this.